Closed-Loop Convex Formulation of Classical and Singular Value Loop Shaping

C. Barratt and S. Boyd

Appeared as a chapter in Control and Dynamical Systems: Digital and Numeric Techniques and Their Applications in Control Systems, C. T. Leondes editor, part 1, 55:1-24, 1993.

We show that control system design via classical loop shaping and singular value loop shaping can be formulated as a closed-loop convex problem. Consequently, loop shaping problems can be solved by efficient numerical methods. In particular, these numerical methods can always determine whether or not there exists a compensator that satisfies a given set of loop shaping specifications. Problems such as maximizing bandwidth subject to given margin and cutoff specifications can be directly solved. Moreover, any other closed-loop convex specifications, such as limits on step-response overshoot, tracking errors, and disturbance rejection, can be simultaneously considered. These observations have two practical ramifications. First, closed-loop convex design methods can be used to synthesize compensators in a framework that is familiar to many control engineers. Second, closed-loop convex design methods can be used to aid the designer using classical loop shaping by computing absolute performance limits against which a classical design can be compared.